Monthly Archives: September 2020

Limits and Continuity: IITJEE Maths: Tutorial Problem Set 2

Problem 1: Find \lim_{x \rightarrow \infty}(1+\frac{2}{x})^{x}.

Problem 2: If G(x)=-\sqrt{25-x^{2}}, then what is the value of \lim_{x \rightarrow 1} \frac{G(x)-G(1)}{x-1}?

Problem 3: If f(x) = (1-x)\tan{\frac{\pi}{2}}, then find the value of \lim_{x \rightarrow 1}f(x)

Problem 4: Find the value of \lim_{x \rightarrow 2}\frac{\sqrt{(x^{2}+5)} -3}{(x-2)}

Problem 5: Find the value of \lim_{x \rightarrow \frac{\pi}{4}}(\sin{2x})^{\tan^{2}{2x}}

Problem 6: Find the value of \lim_{x \rightarrow 0}\frac{\sin{x} - x}{x^{3}}

Problem 7: Find the value of \lim_{n \rightarrow \infty}

Problem 8: If m, n \in N, what is the relationship between m and n if \lim_{x \rightarrow 0} \frac{(\sin)^{n}(x)}{\sin{(x^{m})}}=0

Problem 9: Find \lim_{x \rightarrow 0}\frac{e^{ax}-e^{bx}}{x}

Problem 10: Find the value of a given the following:

f(x) = - 4x when x less than or equal to -2.

f(x)=a.x.x when x greater than -2.

given that \lim_{x \rightarrow -2} {f(x)} exists.

Problem 11: Let f(x) = \frac{\sin{(e^{x-2}-1)}}{\log{(x-1)}}, then find the value of \lim_{x \rightarrow 2} f(x)

Problem 12: Let f(x) = \frac{1}{\sqrt{(18-x*x)}} then find the value of \lim_{x \rightarrow 3}\frac{f(x)-f(3)}{x-3}

Problem 13: The function f(x) = \frac{\log{(1+ax)-\log{(1-ax)}}}{x} is not defined when x is zero. In order to make this function continuous at zero, what should be the value of f at zero?

Problem 14: If the function given below is continuous at x=3, find the value of c:

f(x) = 3x-5 for x<3

f(x)=x+1 for x>3

f(x)=c for x=3

Problem 15: If f(x) = x+2, when x \leq 1 and f(x)=4x-1, when f(x) = 4x-1 when x>1, then which of the following is true ? (a) f(x) is continuous at x=1 (b) \lim_{x \rightarrow 1}f(x) =4 (c) f(x) is discontinuous at x=4 (d) none of these.

Problem 16:

If \phi(x) = \frac{1-\cos{(\lambda x)}}{x \sin{x}}, when x is not zero, and \phi{(0)} = \frac{1}{2}. If \phi{(x)} is continuous at x=0, then find the value of \lambda.

Problem 17:

Let f(x) = \frac{1-\sin{(x)}}{(\pi-2x)^{2}} where x \neq \frac{\pi}{2} and f(\frac{\pi}{2})=\lambda, then which value of \lambda will make f(x) continuous at x = \frac{\pi}{2}

Regards,

Nalin Pithwa

Limits and Continuity: IITJEE Maths : Tutorial problems 1

Problem 1: Which of the following is an indeterminate form ? (a) 1^{1} (b) 0^{1} (c) 1^{0} (d) 0^{0}

Problem 2: Which of the following is not an indeterminate form ? (a) 1^{1} (b) 0 \times \infty (c) 1^{\infty} (d) \infty^{0}

Problem 3: If \lim_{x \rightarrow c} f(x) and \lim_{x \rightarrow c}g(x) exists then which of the following conditions is not always correct ? (i) \lim_{x \rightarrow c}(f(x)+g(x)) = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c}g(x) (ii) \lim_{x \rightarrow c}(f(x)-g(x)) = \lim_{x \rightarrow c}f(x) - \lim_{x \rightarrow c}g(x) (iii) \lim_{x \rightarrow c}(f(x)g(x)) = \lim_{x \rightarrow c}f(x) \times \lim_{x \rightarrow c}g(x) (iv) \lim_{x \rightarrow c} (\frac{f(x)}{g(x)}) = \frac{\lim_{x \rightarrow c}f(x)}{\lim_{x \rightarrow c}g(x)}

Problem 4: If \lim_{x \rightarrow c}(\frac{f(x)}{g(x)}) exists, then (i) both \lim_{x \rightarrow c}f(x) and \lim_{x \rightarrow a}g(x) must exist (ii) \lim_{x \rightarrow a}f(x) need not exist but \lim_{x \rightarrow a}g(x) exists. (iii) neither \lim_{x \rightarrow a}f(x) nor \lim_{x \rightarrow a}g(x) may exist (d) \lim_{x \rightarrow a}f(x) exists but \lim_{x \rightarrow a}g(x) need not exist.

Problem 5: \lim_{x \rightarrow a+}f(x)=l=\lim_{x \rightarrow a-}g(x) and \lim_{x \rightarrow a-}f(x) = m = \lim_{x \rightarrow a+}g(x) then the function (f(x)-g(x)) (i) is continuous at x=a (ii) is not continuous at x=a (iii) has a limit when x \rightarrow a but \lim_{x \rightarrow a}(f(x)-g(x))=l-m (iv) has a limit equal to l-m when x \rightarrow a

Problem 6: If \lim_{x \rightarrow a+}f(x) = l = \lim_{x \rightarrow a-}g(x) and \lim_{x \rightarrow a-}f(x)=m=\lim_{x \rightarrow a+}g(x) then the function (f(x).g(x)) (i) is continuous at x=a (ii) does not have a limit at x=a (iii) has a limit when x \rightarrow a and it is equal to l.m (iv) has a limit when x \rightarrow a but it is not equal to l.m

Problem 7: Find \lim_{x \rightarrow \frac{3.\pi}{4}}\frac{1+\tan{x}}{\cos{(2x)}}

Problem 8: Find \lim_{x \rightarrow e}\frac{\log{x}-1}{x-e}.

Problem 9: Find \lim_{x \rightarrow 0}\frac{a^{x}-b^{x}}{x}

Problem 10: Find \lim_{x \rightarrow 0}\frac{2(1-\cos{x})}{x^{2}}.

Problem 11: Find \lim_{x \rightarrow 0}\frac{\sqrt{1+x}-1}{x}

Problem 12: Find \lim_{x \rightarrow 3-}\frac{|x-3|}{x-3}

Problem 13: Find \lim_{x \rightarrow 0}(\frac{1+\tan{x}}{1+\sin{x}})^{cosec{x}}

Problem 14: Find \lim_{x \rightarrow 0} \frac{(e^{2\sqrt{x}}-1)(\tan{3\sqrt{x}})}{\sin{x}}

Problem 15: Find \lim_{x \rightarrow 0}\frac{\log{\cos{x}}}{x}

Problem 16: Find x if \lim_{x \rightarrow a}\frac{a^{x}-x^{a}}{x^{x}-a^{a}}=-1

Regards,

Nalin Pithwa

Skill Check XV: IITJEE Maths: Foundation: Powers and Roots

Reference: An old ICSE text, class VIII, Oxford Publications, India: Abhijit Mukherjea et al.

Tutorial Exercises:

I) Evaluate \sqrt{38} - \sqrt[3]{38} (you may use a numerical log table)

II) Evaluate 2\sqrt{19} + 3 \sqrt[3]{15} (you may use a numerical log table).

III) Use your calculator to verify: 45^{3}-20^{3}=(45-20)(45^{2}+45 \times 20 +20^{2})

IV) Evaluate \sqrt[3]{980} correct up to three decimal places. You can use a log table.

V) Find the values of the following: use a log table in case you wish:

(5a) 23^{3} (5b) 49^{2} (5c) 28^{2} (5d) 39^{2} (5e) 47^{2} (5f) 19^{3} (5g) 27^{3} (5h) 36^{3} (5i) 41^{3} (5j) \sqrt{15} (5k) \sqrt{26} (5l) \sqrt{29} (5m) \sqrt{37} (5n) \sqrt{47} (5o) \sqrt[3]{13} (5p) \sqrt[3]{31} (5q) \sqrt[3]{22} (5r) \sqrt[3]{50} (5s) \sqrt[3]{34}

VI) Evaluate the following using log tables:

(6a) 19^{2}+10.5^{3} (6b) 45.1^{3}-30.9^{3} (6c) \sqrt{12}+\sqrt{15} (6d) \sqrt{27}-\sqrt{7} (6e) \sqrt[3]{43}+\sqrt[3]{34} (6f) \sqrt[3]{37}-\sqrt[3]{3} (6g) \sqrt{2}+\sqrt[3]{3}-\sqrt[3]{5} (6h) \sqrt[3]{49}+\sqrt[3]{50}-\sqrt{50} (6i) \sqrt{57} (6j) \sqrt{430} (6k) \sqrt[3]{99} (6l) \sqrt[3]{196}

Regards,

Nalin Pithwa

Skill Check: IITJEE XIV: Foundation Maths: Squares and square roots

Tutorial Exercises: (Reference: An old ICSE Text, Oxford Publishers, Class VIII, Abhijit Mukherjea and Sushil Kumar Sharma)

I) Find the value of the following:

(1a) 14^{2} (1b) 27^{2} (1c) 118^{2} (1d) (\frac{1}{5})^{2} (1e) (\frac{3}{8})^{2} (1f) (1\frac{2}{5})^{2} (1g) (2\frac{4}{5})^{2} (1h) (0.1)^{2} (1i) (5.8)^{2} (1j) (3.05)^{2}

II) Which of the following perfect squares will have even square roots and which will have odd square roots?

(2a) 441 (2b) 2916 (2c) 3969 (2d) 21609 (2e) 389376

III) Which of the following numbers are perfect squares ?

(3a) 128 (3b) 256 (3c) 2187 (3d) 6561 (3e) 6084

IV) Find the perfect square obtained by multiplying each of the following numbers by the smallest possible number:

(4a) 882 (4b) 405 (4c) 567 (4d) 1690 (4e) 7776

V) Divide the following numbers by the smallest possible number to make each one a perfect square:

(5a) 1152 (5b) 2187 (5c) 3267 (3d) 1536 (3e) 10140

VI) Find the square roots of the following natural numbers by the prime factorization method:

(6a) 225 (6b) 1296 (6c) 1764 (6d) 2401 (6e) 2916 (6f) 5184 (6g) 7225 (6h) 10816 (6i) 14400 (6j) 13456

VII) Find the square roots of the following fractions by the prime factorisation method:

(7a) \frac{4}{9} (7b) \frac{100}{121} (7c) \frac{49}{64} (7d) \frac{484}{625} (7e) \frac{729}{900} (7f) 1\frac{9}{16} (7g) 1\frac{11}{25} (7h) 1 \frac{32}{49} (7i) 10 \frac{9}{16} (7j) 8\frac{1}{36}

VIII) Find the square roots of the following decimals by the prime factorization method:

(8a) 0.0001 (8b) 1.44 (8c) 3.24 (8d) 7.29 (8e) 10.24 (8f) 30.25 (8g) 0.0049 (8h) 0.1296 (8i) 15.21 (8j) 0.2025

IX) Find the square roots of the following numbers by the division method:

(9a) 4624 (9b) 11664 (9c) 16641 (9d) 44521 (9e) 426409 (9f) 550564 (9g) 840889 (9h) 917764 (9i) 1014049 (9j) 1560001

X) Find the square roots of the following decimal fractions by the division method:

(10a) 34.81 (10b) 44.89 (10c) 5.4756 (10d) 1.3225 (10e) 4.9284 (10f) 11.2896 (10g) 4.5796 (10h)36.2404 (10i) 1.304164 (10j) 4.609609

XI) Find the square roots of the following common fractions by the division method:

(11a) \frac{121}{289} (11b) 1\frac{168}{361} (11c) \frac{676}{729} (11d) \frac{1681}{2209} (11e) \frac{5041}{7921}

XII) Find the square roots of the following numbers, correct up to 3 decimal places:

(12a) 2 (12b) 3 (12c) 5 (12d) 11 (12e) 13 (12f) 1458 (12g) 27196 (12h) 14502 (12i) 1828 (12j) 146923 (12k) 1.6 (12l) 1.09 (12m) 2.87 (12n) 3.99 (12o) 21.654 (12p) \frac{1}{5} (12q) 1 \frac{3}{4} (12r) \frac{7}{20} (12s) \frac{7}{8} (12t) 1\frac{7}{16}

XIII) Find the smallest number that needs to be subtracted from 66671 in order to get a perfect square.

XIV) Find the smallest number that needs to be subtracted from 1051149 in order to get a perfect square.

XV) Find the smallest number that needs to be added to 1485155 in order to get a perfect square.

XVI) Find the smallest and greatest 5-digit numbers that are perfect squares.

XVII) The area of a square is 2.815684 square cm. Find its length.

XVIII) A farmer has 21126 seedlings, which he intends to plant in an equal number of rows and columns. What is the maximum number of rows of seedlings that can be planted? How many seedlings will be left unplanted?

XIX) Chairs need to be laid out in an equal number of rows and columns. If there are 1817 chairs in the stadium, find at least how many more chairs need to be brought in.

Regards,

Nalin Pithwa

Skill Check XIII IITJEE Foundation Maths: Approximation

Introduction:

(Reference: an old ICSE grade VIII text book, New Guided Mathematics, class 8, Abhijit Mukerjea and Sushil Kumar Sharma, Oxford University Press, India)

Rahul needs to divide a web page into three vertical parts. He wants it to look neat and thus wants the measurements to be perfect. But the web page is 640 pixels wide and 640/3 = 213.3 pixels. Is it necessary for Rahul to be so accurate? If he rounds off each part to 213 pixels , his page can be divided into 3 parts measuring 213, 214, and 213 pixels. Although the parts are not exactly equal, they are approximately equal to each other, a relationship that is denoted by the symbol \approx.

Rounding Off Decimal Fractions

Rounding off a decimal fraction to its decimal places:

  1. If the digit in the (n+1)th decimal place is less than 5, then the digit in the (n+1)th decimal place, and all the digits after it, are simply omitted.
  2. If the digit in the (n+1)th decimal place is 5 or more than 5, the nth digit is increased by 1 and the digit in the (n+1)th decimal place, and all the digits are it, are omitted.

Example 1: Convert \frac{0}{7} into a decimal fraction and round off your answer to:

i) 1 decimal place

ii) 3 decimal places

iii) 5 decimal places

iv) a whole number

Solution: \frac{9}{7}=1.2857142\ldots

(i) In order to round off the answer to the tenths or 1 decimal place, we consider the (1+1)th decimal place digit. As 8>5, the digit in the tenths place is increased by 1 and all the digits after it are omitted. Thus, \frac{9}{7}=1.3

(ii) As the (3+1)th decimal place digit is 7 and 7>5, the digit in the thousandths place or 5 is increased by 1 and all the digits after it are omitted. Thus, \frac{9}{7}=1.286

(iii) As the digit in the (5+1)th decimal place is 4 and 4<5, the digit in the (5+1)th decimal place and all the digits after it are omitted. Thus, \frac{9}{7}=1.28571

(iv) As the digit in the tenths place is 2 and 2<5, it is omitted and all the digits after it are also omitted. Thus, \frac{9}{7}=1

TRY THIS: Convert \frac{10}{20} into a decimal fraction and round off your answer to 4 decimal places.

Rounding Off to a Specified Unit:

We have just learned how to round off our answers to a specified number of decimal places. But in real life we have to use our sense and figure out how much to round off such that our answer makes sense.

  1. Raima buys a pair of sunglasses marked at Rs. 690.75 selling at a discount of 35 %. The shopkeeper charges her Rs. 449. Should Raima complain that she has been charged Rs. 0.0125 more?
  2. A lady wishing to buy a gold ring asks the jeweller the market price of gold. Would it make more sense to the lady if the jeweller quoted the price of gold in kilograms or grams?
  3. A traveller consults a railway timetable to find out how much it would cost to travel from Kolkata to Mumbai. Would it help the traveller if the fare chart quotes fares in paise per meter rather than rupees per kilometer?

Example 2: An ice-cream vendor buys 60 ice-creams for Rs. 210 and wishes to earn a 40% profit. At what price should he sell each ice-cream?

Answer 2: Cost price per ice cream is 210/60=Rs. 3.50

Profit expected per ice-cream is 3.5 x 40/100, that is, Rs. 1.40

Accurate SP per ice-cream is Rs. 3.50 + Rs. 1.40 that is Rs. 4.90

As it would be difficult for all his customers to pay exact change (Rs. 4.90), the vendro could round it off to a reasonable price of Rs. 5 per ice-cream.

Example 3: What is the capacity of a cubical ink pot that is 0.04 m long?

Answer 3: The calculation of capacity with length in meters would give an answer in kilolitres. But the vessel in question is an ink pot and not a tanker !! Thus, it would make more sense to convert the length to centimeters and find the capacity in cubic cm or millilitres.

0.04 m is 4 cm.

Capacity of ink pot is 4 x 4 x 4, that is, 64 cubic cm.

Significant Digits in Decimals

We know that 00002.3=2.3=2.30000

The significant or meaningful digits in the above numbers are only the digits 2 and 3 on either side of the decimal point that give us an idea of its value and location on the number line.

  1. All non-zero digits in a decimal number are significant digits.

(a) 5.695 has four significant digits.

(b) 58.2 has three significant digits.

2. The zeroes between non-zero digits in a decimal number are significant digits.

(a) 3.01 has three significant digits, being 3, 0 and 1.

(b) 5.2001 has five significant digits, being 5, 2, 0, 0 and 1.

3. The zeros to the left of the first non-zero digit are not significant digits.

(a) 0.09 has only one significant digit, being 9.

(b) 0.00012 has only two significant digits, being 1 and 2.

4. The zeros to the right of the last non-zero digit may or may not be significant. The condition depends on the unit of measurement or the need for approximation.

Case I: When zeros to the right of the last non-zero digits are significant.

Example : Convert 1.2000 m into centimeter.

Solution: As 1.2000m is 120 cm, so in 1.2000 m there are three significant digits, being 1, 2 and 0.

Example: Convert 3.6000000 kg into mg.

Solution: As 3.6000000 kg is 3600000 mg, in 3.6000000 kg, there are 7 significant digits being 5, 6, and give zeros.

Case II: When zeros to the right of the last non-zero digit are not significant.

Example: Express Rs. 3.60000 in paise.

Solution: As Rs. 3.60000 is 360 paise, Rs. 3.60000 has only three significant digits, being 3, 6 and 0, the three zeros after it being insignificant.

Example: Given the area of the Arctic Ocean as 13079000 square km, express its area in thousand square km.

Solution: 13079000 sq km x 1/1000 is 13079 thousand sq km. Thus, 13079000 sq km has only 5 significant digits, being 1, 3, 0, 7 and 9, the three zeros after it being insignificant.

Approximation to Signficant Digits

Example 1: Approximate 0.0003801 to 1 significant digit.

Solution 1: The given number has 4 significant digits, being 3, 8, 0 and 1. To approximate it to 1 significant digit, it will have to be rounded off to its ten-thousandths place where its first significant digit is. As 8 in the hundred-thousandths place is greater than 5, 0.0003801 is 0.0004.

Example 2: Approximate 3.1428571 to 3 significant digits.

Solution 2: The given number has 8 significant digits. To approximate it to 3 significant digits, it will have to be rounded off to the hundredths place. As 2 in the thousandths place is less than 5, 3.1428571 \approx 3.14

Representation of Numbers

Let us recall how we expressed a decimal number in expanded form in previous classes:

Example : Write 5873.1264 in expanded form:

Solution: 5873.1264 can be written as

5000+800+70+3+0.1+0.02+0.006+0.0004 or

5 \times 10^{3} + 8 \times 10^{2} + 7 \times 10^{1}+ 3 \times 10^{0}+ 1 \times 10^{-1} + 2 \times 10^{-2} + 6 \times 10^{-3} + 4 \times 10^{-4}

TRY THIS!

Write 5840.183 in expanded form.

Thus, whatever be the place of a digit in the integral or decimal part of a number, its place value can be described as a multiple of a power of 10.

The distance between Earth and Pluto is 575,00,00,000 km. This is too big a number to communicate and remember:

(i) Thus, it is approximated to its significant digits and represented as 5 billion 750 million km. Or as 575 crore km OR

(ii) All the significant digits are considered and it is written in scientific notation as the product of a decimal number less than 10 and a power of 10, that is, 5.8 \times 10^{9} km.

We know that a millilitre is one-thousandth part of a metre. Scientists and industrialists are very excited, about “nanotechnology” now-a-days. Do you know the relationship between a nanometre and a metre?

1 nanometre is 0.000000001 of a metre or a nanometre is a thousand-millionth part of a metre. Such a small decimal number is again written in scientific notation as the product of a decimal number less than 10 and a power of 10 as follows: 1 nm = 1 \times 10^{-9} m.

Example: Write 0.000200100 in scientific notation.

Solution: The significant digits in the number are 2, 0, 0, and 1. The decimal number less than 10 is 2.001 and as a product of power of 10, the scientific notation for 0.000200100 is 2.001 \times 10^{-4}.

Try to figure out the weight of an electron if it is given in scientific notation as equal to 9.10908 \times 10^{-31}kg.

Exercises:

  1. Approximate 0.0004567 to 1 significant digit.
  2. Write 6245.3173 in expanded form.
  3. Round off 22.5% of Rs. 58.50 to the nearest piase.
  4. Express 7 liters 54 ml to the nearest litre.
  5. Write 7.000 40 23587 in scientific notation.

Regards,

Nalin Pithwa

To balance, to focus, for inner calm and concentration

Happiness Chemicals and how to hack them:

I) DOPAMINE: The reward chemical:

  • Completing a task
  • Doing self-care activities
  • Eating food
  • Celebrating little wins

II) SEROTONIN: The mood stabilizer:

  • Meditating
  • Running
  • Sun exposure
  • Walk in nature
  • Swimming
  • Cycling

III) OXYTOCIN: The love hormone

  • Playing with a dog
  • Playing with a baby
  • Holding hand
  • Hugging your family
  • Give compliment

IV) ENDORPHIN: The pain killer

  • Laughter exercise
  • Essential oils
  • Watch a comedy
  • Dark chocolate
  • Exercising

The City with a large heart : Bombay/Mumbai

Mumbai is rich,

Mumbai is poor,

Mumbai is fast,

Mumbai is slow.

Little bit sweet,

and little bit sour.

Sometimes its hot,

but not too cold.

Mornings r energetic

& evenings are electric.

Noons r lazy but

Nights are crazy.

And anyone U ask,

he always says “M busy”

Dude, life in Mumbai

Is not so easy…!

There is lot of Masti with

little bit of Maska…

Welcome to the city that

can’t live without our Bollywood Chaska!

Sev puri, Vada Pav and bhel puri

are all Mumbai chaat.

Relishing it with spicy chutney is no easy art

From popcorn to icecream, all sold

on cart

Mumbai o Mumbai you’re always

close to my heart!

Where local trains

usually run on time

And violently rushing

for a seat is not a crime.

Here 3 PM for lunch and

12AM to dine,

People face hardships,

but still say, “It’s fine”!

From Siddhivinayak in Dadar

to Woodhouse Cathedral in Town,

And ISKCON in Juhu to Haji Ali in

Mumbai’s Crown,

Marathi, Malayalee, Christian to

Gujarati,

Everyone together celebrate

Christmas and Diwali,

Holi is colourful and Diwali is cheerful,

Spend some time here and your life

will be unforgetful!

Billionaires to beggars,

all found in this city,

Be careful dude,

this place is a bit witty…

Overall this dream world

is huge but pretty

Mumbai Mumbai,

you’re wonderful city

After all a mother is called

MUM in English

BA in Gujarati &

Ai in Marathi

That’s My Mumbai

Dedicated to all Mumbaikars !!